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Chapter 10: Problem 30

To derive the ideal-gas equation, we assume that the volume of the gas atoms/molecules can be neglected. Given the atomic radius of neon, \(0.69 \AA\), and knowing that a sphere has a volume of \(4 \pi \mathrm{r}^{3} / 3\), calculate the fraction of space that Ne atoms occupy in a sample of neon at STP.

Short Answer

Expert verified
The fraction of space occupied by Ne atoms in a sample of neon at STP can be calculated using the following steps: 1. Calculate the volume of a single Ne atom: \(V_{atom} = \dfrac{4}{3} \pi (0.69 \times 10^{-10})^3\) 2. Calculate the volume of 1 mole of Ne gas at STP: \(V_{stp} = 22.4 \times 10^{-3} m^3\) 3. Find the number of atoms in 1 mole of Ne: \(N_{atoms} = 6.022 \times 10^{23}\) 4. Compute the total volume of Ne atoms in 1 mole: \(V_{total} = V_{atom} \times N_{atoms}\) 5. Calculate the fraction of space that Ne atoms occupy in the sample at STP: \(Fraction_{occupied} = \dfrac{V_{total}}{V_{stp}}\) By plugging in the values calculated in previous steps and simplifying the expression, we can find the fraction of space occupied by Ne atoms.

Step by step solution

01

Calculate the volume of a single Neon atom

We know that the atomic radius of neon is \(0.69 \times 10^{-10}\) meters and a sphere has a volume of \(\dfrac{4}{3} \pi r^{3}\). So, we can compute the volume of a single Neon atom as follows: \[V_{atom} = \dfrac{4}{3} \pi (0.69 \times 10^{-10})^3\]
02

Calculate the volume of 1 mole of Neon gas at STP

At Standard Temperature and Pressure (STP), 1 mole of any ideal gas occupies \(22.4\) liters of volume. So, we can calculate the volume of 1 mole of Neon at STP: \[V_{stp} = 22.4 L = 22.4 \times 10^{-3} m^3\]
03

Find the number of atoms in 1 mole of Neon

In one mole of a substance, there are Avogadro's number (\(N_A\)) of particles (atoms, molecules, or ions). Avogadro's number is approximately \(6.022 \times 10^{23}\) particles per mole. Thus, in 1 mole of Neon, there are: \[N_{atoms} = N_A = 6.022 \times 10^{23}\]
04

Compute the total volume of Neon atoms in 1 mole

To find the total volume occupied by the Neon atoms in 1 mole, we will multiply the volume of a single atom (calculated in step 1) by the number of atoms in 1 mole (calculated in step 3): \[V_{total} = V_{atom} \times N_{atoms}\]
05

Calculate the fraction of space that Ne atoms occupy in the sample at STP

Now, we can calculate the fraction of space that Ne atoms occupy in the sample at STP by dividing the total volume occupied by Neon atoms (calculated in step 4) by the volume of 1 mole of Neon gas (calculated in step 2): \[Fraction_{occupied} = \dfrac{V_{total}}{V_{stp}}\] Calculate the fraction by plugging in the values calculated in previous steps and simplifying the expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atomic Radius
The atomic radius is the distance from the nucleus of an atom to the outer boundary of its electron cloud. This value helps us understand the size of individual atoms. Neon has an atomic radius of \(0.69 \text{ 脜}\), which is equivalent to \(0.69 \times 10^{-10}\) meters.
Knowing the atomic radius is essential when calculating the volume of an atom, especially since many physical and chemical properties are influenced by atomic size. For neon, with its noble gas status, the atomic radius determines how it occupies space at the molecular level.
Avogadro's Number
Avogadro's number is a fundamental constant used in chemistry to denote the number of particles in a mole of a substance. It is approximately \(6.022 \times 10^{23}\).
This number helps us convert between atoms/molecules and moles, making it easier to calculate quantities in chemical reactions and processes. When dealing with gases like Neon, knowing Avogadro's number allows us to find out how many neon atoms are in a sample.
  • Useful for stoichiometry in chemistry.
  • Allows transition between atomic scale and macroscopic measurements.
Volume of a Sphere
The formula to calculate the volume of a sphere is \( \dfrac{4}{3} \pi r^3 \). This formula helps determine how much space a spherical object, like an atom, occupies.
Given the atomic radius of neon, calculating a single atom's volume is crucial in understanding how much of a given volume of gas is actually filled by matter.
  • Important in derivations involving gases and molecules.
  • Helps visualize how much physical space atoms take.
Using the formula, the volume of a single neon atom is determined, which then contributes to understanding collective atomic arrangements in gases.
Standard Temperature and Pressure
Standard Temperature and Pressure (STP) is a reference point in chemistry used to define conditions of \(0^\circ\text{C}\) (273.15 K) and 1 atmosphere of pressure. At STP, one mole of an ideal gas occupies 22.4 liters of volume.
This standard is crucial when performing calculations involving gas volumes, because it provides a benchmark that relates to the ideal gas law.
  • Assists in comparing gas behaviors under standard conditions.
  • Facilitates calculations in varied scientific disciplines.
Understanding STP allows us to accurately measure and predict how gases like neon behave under these conditions.

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Most popular questions from this chapter

(a) Amonton's law expresses the relationship between pressure and temperature. Use Charles's law and Boyle's law to derive the proportionality relationship between \(P\) and \(T\). (b) If a car tire is filled to a pressure of \(32.0 \mathrm{lbs} / \mathrm{in}^{2}\) (psi) measured at \(75^{\circ} \mathrm{F}\), what will be the tire pressure if the tires heat up to \(120^{\circ} \mathrm{F}\) during driving?

Consider a mixture of two gases, \(A\) and \(B\), confined in a closed vessel. A quantity of a third gas, \(C\), is added to the same vessel at the same temperature. How does the addition of gas \(C\) affect the following: (a) the partial pressure of gas A, (b) the total pressure in the vessel, (c) the mole fraction of gas B?

Which of the following statements best explains why a closed balloon filled with helium gas rises in air? (a) Helium is a monatomic gas, whereas nearly all the molecules that make up air, such as nitrogen and oxygen, are diatomic. (b) The average speed of helium atoms is greater than the average speed of air molecules, and the greater speed of collisions with the balloon walls propels the balloon upward. (c) Because the helium atoms are of lower mass than the average air molecule, the helium gas is less dense than air. The mass of the balloon is thus less than the mass of the air displaced by its volume. (d) Because helium has a lower molar mass than the average air molecule, the helium atoms are in faster motion. This means that the temperature of the helium is greater than the air temperature. Hot gases tend to rise.

(a) What conditions are represented by the abbreviation STP? (b) What is the molar volume of an ideal gas at STP? (c) Room temperature is often assumed to be \(25^{\circ} \mathrm{C}\). Calculate the molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and 1 atm pressure. (d) If you measure pressure in bars instead of atmospheres, calculate the corresponding value of \(R\) in L-bar/mol-K.

You have a gas at \(25^{\circ} \mathrm{C}\) confined to a cylinder with a movable piston. Which of the following actions would double the gas pressure? (a) Lifting up on the piston to double the volume while keeping the temperature constant; (b) Heating the gas so that its temperature rises from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\). while keeping the volume constant; (c) Pushing down on the piston to halve the volume while keeping the temperature constant. ll be produced?
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